How do I describe a region that is not a box?

Let the bounds lean on the variables outside them. With the integrand 1, x from 0 to 1-y-z, y from 0 to 1-z and z from 0 to 1 describe the corner of the unit cube cut off by the plane x + y + z = 1, and the page gives 1/6, its volume.

Can I leave the bounds empty?

Yes: clear all six for the iterated antiderivative, and x*y*z comes back as x²*y²*z²/8. No + C is printed, because what is left undetermined is a function of each variable rather than one constant.

What happens when the library's value is wrong?

When the sampling can tell, it is not shown. Once the library answers, the page samples the region at 20 and then 40 points per variable. If those two samplings do not settle, the result reads Does not settle; if the library's figure is further from the settled sampling than the sampling's own error, it reads Not shown, and the numbers that disagreed are printed in its place. A numerical value that agrees is shown only to the digits the sampling confirms. When the region cannot be sampled, the page says the figure rests on the library alone.

Triple Integral Calculator

Opens the integral calculator with three variables, on x*y*z over a box that comes to 9/2. The inner integral runs first, then the middle, then the outer, so an inner bound may use the variables outside it, and a value the sampling of the region contradicts is not shown.

IntegralWorking∭ x*y*z dx dy dz, x from 0 to 1, y from 0 to 2, z from 0 to 3 · read as x*y*z · symbolic

Integrand

Method

Integrals

Result: Working
Inner variable
Inner bounds
Numeric: n/a

Middle variable

Middle bounds

Outer variable

Outer bounds

The inner integral runs first, then the middle, then the outer. Inner bounds may use the middle and outer variables, middle bounds may use the outer one, and the outer bounds take numbers only.

x^2 dxindefinite
x³/3 + C
x^2 from 0 to 3definite
9
exp(x^2) dxno elementary form
named, not approximated

Accuracy. Exact symbolic integration for the functions it supports, with the expression it actually read printed back so you can check the parse. An integral with no elementary antiderivative is named as such rather than approximated silently, and no step-by-step derivation is claimed.

What you can type

Numbers, single letters for variables, the constants pi and e, the signs + - * / ^ and brackets, and these functions: sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, exp, sqrt, abs, log. ln is read as log, which is the natural logarithm here, and arcsin, arccos and arctan are read as asin, acos and atan. Multiplication can be left out where it is obvious, so 2x, 3 sin(x) and 2(x + 1) all read the way you would write them on paper, and the stage prints back what that came to. Leave both bounds empty for an antiderivative, or fill them in for a definite integral. Powers go up to 64, and the expression can be up to 500 characters long.

Common questions

How do I describe a region that is not a box?: Let the bounds lean on the variables outside them. With the integrand 1, x from 0 to 1-y-z, y from 0 to 1-z and z from 0 to 1 describe the corner of the unit cube cut off by the plane x + y + z = 1, and the page gives 1/6, its volume.
Can I leave the bounds empty?: Yes: clear all six for the iterated antiderivative, and x*y*z comes back as x²*y²*z²/8. No + C is printed, because what is left undetermined is a function of each variable rather than one constant.
What happens when the library's value is wrong?: When the sampling can tell, it is not shown. Once the library answers, the page samples the region at 20 and then 40 points per variable. If those two samplings do not settle, the result reads Does not settle; if the library's figure is further from the settled sampling than the sampling's own error, it reads Not shown, and the numbers that disagreed are printed in its place. A numerical value that agrees is shown only to the digits the sampling confirms. When the region cannot be sampled, the page says the figure rests on the library alone.

Exact symbolic integration for the functions it supports, with the expression it actually read printed back so you can check the parse. An integral with no elementary antiderivative is named as such rather than approximated silently, and no step-by-step derivation is claimed.